What is the ratio of the gravitational pull of the sun on the moon to that of the earth on the m

What is the ratio of the gravitational pull of the sun on...

Key Concepts

Spheres of Influence

The concept of a sphere of influence refers to the region around a celestial body within which it exerts a dominant gravitational pull over its satellites. Even though the Sun’s gravitational force on the Moon may be larger in magnitude compared to Earth’s force, the Moon remains under the Earth's gravitational control because it is within Earth’s sphere of influence, which explains why the Moon is conceptually regarded as an Earth-orbiting satellite.

Gravitational Force Ratio

The gravitational force ratio involves comparing the gravitational pulls exerted by two different bodies on a third object. By considering the masses of the bodies and the distances involved, one can determine which gravitational force is dominant. This ratio is essential in analyzing multi-body systems, such as comparing the Sun's and Earth's gravitational effects on the Moon.

Orbital Mechanics

Orbital mechanics is the study of the motions of objects under the influence of gravitational forces. It encompasses the dynamics of orbits, the effects of multiple gravitational influences, and the determination of dominant gravitational interactions. This field helps explain why, even if the Sun's pull on the Moon can be stronger in absolute terms, the Moon is generally thought of as orbiting the Earth due to its proximity and the gravitational interplay governing its motion.

Newton's Law of Universal Gravitation

Newton's Law of Universal Gravitation states that every two masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. This principle provides the foundation for calculating the gravitational forces exerted between celestial bodies such as the Earth, Moon, and Sun.

Inverse Square Law

The inverse square law is a key aspect of gravitational interactions, explaining that the strength of the gravitational force diminishes in proportion to the square of the distance between two objects. This concept is crucial for understanding why gravitational influences become significantly weaker as the separation between objects increases.

Transcript

00:03 So in this problem, we're asked to find the ratio of the gravitational force that the sun applies to the moon, divided by the force that the earth applies on the moon.

00:12 So for any two objects, objects one and two, the gravitational force between them is just going to be g, the gravitational constant, times the two masses, divided by the square of the distance between the two masses.

00:26 So to find the ratio here, we're just going to apply that equation to both the sun, the moon, and the earth in the moon.

00:33 In both cases, then, we're going to have just g, and then times the mass of the sun, times the mass of the moon, divided by the distance between the sun and the moon squared.

00:46 So that's the force the sun applies in the moon.

00:49 And then to get the ratio, now we divide this by the force the earth puts on the moon.

00:55 So in the denominator now, we have g again, mass of the earth times mass of the moon, and divided by the distance between the earth and the moon squared.

01:08 And so now when we're evaluating this, the gravitational constant cancels out.

01:13 So the ratio doesn't actually depend on the gravitational constant at all.

01:17 And then also the mass of the moon cancels out as well.

01:20 So the only things that matters here for this ratio are the mass of the sun, the mass of the earth, and the distance between the sun and the moon and the earth and the moon.

01:27 So this then reduces to just the mass of the sun.

01:32 Divided by the mass of the earth, and then multiplied by the distance between the earth and the moon, and the distance between the sun and the moon, and then both of those are squared.

01:47 So now we look up the values for those in the appendix...

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